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tags: Paradigm RFQ
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# Basis Basics - Part 2
Max Holloway | [@maxholloway](https://twitter.com/maxholloway)

*Here we pick up where [Basis Basics - Part 1](https://www.paradigm.co/blog/basis-basics-pt-1) left off by discussing advanced basis trading strategies. We introduce the concept of leveraged basis trading, and we provide a framework for deciding between lending and basis trading on both fixed-expiry and perpetual futures.*

## Basis Background
In [part 1](https://www.paradigm.co/blog/basis-basics-pt-1) of the series, we defined a spot-futures basis $b$ as the difference between a futures price $p_f$ and a spot price $p_s$. We also defined the basis percentage, $b_\%$.
\begin{align*}
b & = p_f - p_s \\
b_\% & = \frac{p_f - p_s}{p_s}.
\end{align*}

We showed that when the basis is positive, traders can receive a fixed interest rate by shorting futures and buying spot asset, i.e. shorting the basis.

*Basis percentage for Ethereum's June 24, 2022 futures contract on FTX. Spot price comes from FTX's ETH/USD spot market. Source: [FTX's API](https://docs.ftx.com/#overview).*

Here, we expand our analysis to leveraged basis trades and perp funding rate trades, and find conditions for which a trader should trade the basis rather than lend USD.

## Leveraged Basis Trading
### Leveraged basis trading returns
Basis trades offer a fixed-interest payoff structure and provide a market-neutral profit opportunity. Due to the basis' low risk, it is worthwhile to ask: is there a *lever* through which traders can increase the trade's return while also increasing the trade's risk? It turns out there is: leveraged basis trading through spot margin lending.

Consider an example of going short the basis when the basis is positive (i.e. when p_f > p_s$). Returns with an unleveraged portfolio with \$100 would be \$$b_\%$. Now suppose that those \$100 were used to go long \$500 worth of the risky asset `A` via margin loans (buy `A`, deposit `A` as collateral for a margin loan of USD, and repeat until \$400 USD have been borrowed). Now let the interest rate on a loan held until futures expiry received on `A` deposits be $r_A$, while the interest rate charged on USD loans be $r_{USD}$; note these are not annual interest rates, but rather percentage interest charged for holding until the futures contract expires. Then the net interest rate paid when achieving the leverage ratio $l$ is
\begin{align*}
r_{net} & = (l-1) \cdot r_{USD} - l \cdot r_A = l(r_{USD} - r_A) - r_{USD}.
\end{align*}

In most cases, such as with FTX's lending platform, the asset `A` used as collateral does not accrue interest while being used as collateral[^1]. In those cases, $r_A=0$. On the other hand, on-chain money markets like Aave and Compound *do* allow borrowers to earn interest on the tokens deposited as collateral[^2] [^3].

In general, the return, $r_{trade}$, achieved by the leveraged basis trade is given by subtracting the cost of leverage from the basis percentage as follows.

\begin{align*}
r_{trade} = l \cdot b_{\%} - r_{net} = l(b_{\%} - r_{USD} + r_A) + r_{USD}.
\end{align*}

This gives the proportion return at the expiry of the futures contract. If the trader has access to low USD interest rates relative to the basis, and they also do not receive interest on their collateral, then their returns from basis trade can be approximated as

\begin{align*}
r_{trade-no-interest-rates} \approxeq l \cdot b\%.
\end{align*}

When USD interest rates are low, there exists a trade with known payoff whose returns can be amplified with USD loans. However, although the payoff at futures expiry is known, there is still drawdown risk of holding the leveraged basis position, which we discuss now.

### Leveraged basis trading risks
Of course, as leverage increases, so does the risk of basis trading. Unlike familiar derivative instruments like futures and options, the payoff function for basis trades has no direct relationship to the price changes of a single underlying asset. Instead, the basis trader faces risk of the *difference* between futures and spot prices going against their favor. It is guaranteed that the basis will go to zero at the time of the futures contract expiry. However, there is no guarantee that the basis will monotonically approach 0 as the futures contract expiry approaches. This can be best understood with an example.

Suppose the basis is positive, and a trader goes short the basis with 10x leverage. Suppose the basis is $5\%$ when they enter the trade, and the basis reaches $10\%$ while the trade is still open. Then, at that moment, the trader would have PnL
\begin{align*}
(-10) \cdot (10\% - 5\%) = -50\%.
\end{align*}
Even worse, if the basis reaches $15\%$, then the trader would have PnL of $-100\%$, being liquidatated unless they inject more capital into the strategy. In fact, the collateral requirements on the futures position and the margin lending position would likely lead to a margin call far before the basis reached $-15\%$.

It is worth noting that historically, due to this trade's pervasiveness, basis risk has been quite low. The following graph is meant to provide a ballpark for the order of magnitude that a basis might change over the course of approximately 6 months.

*The maximum change in basis percentage during the lifetime of the 06/24 futures contract for various assets. This represents the maximum percentage loss that a basis speculator could incur by speculating with 1x leverage on a basis trade. Source: [FTX's API](https://docs.ftx.com/#overview).*

In this graph, we plot the maximum change that the basis incurred for a number of FTX's popular 06/24 futures contracts. This is an upper bound on the basis risk that a 1x leveraged trader would face if they were trading a basis for one of these coins. No 1x leveraged traders would be at risk of liquidation. However, if they were highly leveraged on their positions, e.g. by 10x or 20x, then the basis risk could certainly be a legitimate risk when conducting this trade.

Although basis trades have a fixed return at the expiry of the futures contract, there is no bound on the trader's drawdown between the time that the basis trade is opened and the futures expiry. This drawdown can enough to cause an account to be liquidated, and the more leveraged the basis trade is, the more likely an account will be liquidated due to basis risk.

## Basis <> Lending Arbitrages
We now turn our attention to framing basis trading in relation to lending. Specifically, we compare two trades -- shorting the basis and providing USD loans -- and we provide guidance for traders to decide between these two yield-bearing opportunities.

### Lending arbitrage with fixed-expiry futures

It is tempting to compare fixed-expiry futures contracts to risk-free fixed-interest lending markets. In both cases, there is a specific time in the future when the trader is entitled to a fixed return on their capital investment. However, there are three key distinctions between shorting the basis and loaning out USD.

First, fixed-interest lenders have no notion of liquidation risk[^4]; they simply provide capital. This is in contrast with a basis trade, where the trader's collateral value may temporarily decrease before the futures expiry. In practice, these intermediate basis fluctuations only affect highly leveraged basis traders. Still, this makes lending less risky than basis trading.

Second, since the basis may change over time, it is possible for a trader to be receive their payout before the expiry of the contract. For instance, suppose the basis percentage is $3%$, and a trader goes short the basis 90 days before the futures contract expiry. Now suppose that one day after the trader originates their position, the basis fluctuates down to $0\%$. The trader can now cash in their $3\%$ return after locking capital for a single day. In comparison, if they provided a $3\%$ fixed-rate loan for 90 days, they would need to hold that loan until expiry. This example illustrates the fact that basis trading can yield its fixed-rate returns *before* the futures expiry date, which makes basis trading less capital intensive than fixed-rate lending in expectation.

Third, it should go without saying that there are different risk profiles for lending and basis trading. However if we assume that both strategies are running through a centralized exchange, such as FTX, then the risk profile may be similar for both assets; importantly, counterparty/credit risk is beyond the scope of this article, and we urge readers to do their own research on those risks.

Although both basis trading and fixed interest lending give guarantees on the returns at a fixed date in the future, basis trading entails more intermediate risk than fixed interest lending, while also providing potential for faster payoffs. In practice, we should expect the prospect for faster payoffs to outweigh the basis risk, due to the fact that basis risk is so small and the potential for faster payoffs is so significant.

Therefore, if the basis percentage is positive, all but the most risk-averse traders should short the basis with fixed return $b\%$ rather than lend USD at a fixed rate of $b\%$ or lower[^5].

### Lending arbitrage with perpetual futures
Perhaps even simpler than the arbitrage analysis between fixed-expiry futures and fixed-rate lending is the comparison between perpetual futures (perps) and money markets with variable interest rates.

Perps maintain their price parity to spot assets through their *funding rate*: an amount of USD (or USD-equivalent stablecoin) to be transferred between longs and shorts over time; if the perp price is above the spot price, then the funding rate is positive, and longs pay shorts; if the perp price is below the spot price, then the funding rate is negative, and shorts pay longs. This mechanism is a form of control (namely, proportional control[^6]) that keeps the perp price near the spot price.

Suppose the perp funding rate for an asset `A` is very positive. Then traders can receive payment for shorting the perp. If the trader wants to do so without assuming price risk of `A`, they can buy the spot asset at the same time they short the perp, earning the perp funding rate with negligible price exposure. Voila! We have replicated the fixed-expiry short basis trade with perpetuals! If the trader wishes to achieve leverage on their trading, they can lever up their positive exposure to the spot token through lending markets. Similar to before, if we let $l$ be the spot leverage ratio, $r_A$ be the interest rate from lending asset `A`, $r_{USD}$ the interest rate paid on USD loans, $r_{net}$ be the net interest rate cost from levering up spot exposure via lending, and $r_f$ be the funding rate of the perp, the return from this trade would be

\begin{align*}
r_{perp-trade} = l(r_f) - r_{net} = l(r_f + r_A - r_{USD}) + r_{USD}.
\end{align*}

While this formula is eerily similar to the one for returns from a fixed-term futures basis trade, the $r_*$ terms in this formula are not fixed, and instead they are random variables. Therefore, this trade is not a pure arbitrage, but instead is a statistical arbitrage: traders can use distributional assumptions about the $r_*$ terms to calculate a distribution of $r_{perp-trade}$.

If we were to assume that for some perp return ratio, $r^*_{perp-trade}$, traders will be indifferent between basis trading and lending, then we would have that
\begin{align*}
r^*_{perp-trade} &= r_{USD} \text{, so} \\
l^*(r_f^* + r_A^* - r_{USD}^*) + r_{USD} & = r_{USD} \text{, so} \\
l^*(r_f^* + r_A^* - r_{USD}^*) & = 0.
\end{align*}

Thus, $r_f^* = r_{USD}^* - r_A^*$.

This expression relates the arbitrage-free perp funding rate to the arbitrage-free variable interest rates of USD and asset `A`. When $r_f > r_{USD} - r_A$, risk-neutral traders should short perps and go long spot; when $r_f < r_{USD} - r_A$, risk-neutral traders should lend USD instead of performing a funding rate trade.

Here is an example to illustrate. Suppose a trader expects that the variable interest rate they will receive for lending USD for a month is $1\%$, and they expect they will receive $2\%$ in funding rate payments by purchasing spot and shorting perps on some crypto-asset `A`. Furthermore, assume that they expect the interest earned for lending asset `A` to be $0\%$ (as is the case when providing `A` as collateral on FTX). Then we have that $r_f = 2\%$, $r_{USD} = 1\%$, and $r_A = 0\%$. Thus, we have that $r_f > r_{USD} - r_A$, so the risk-neutral trader should pursue the funding rate trade.

Unfortunately, applying the perp no-arbitrage condition in practice suffers from two factors that we gloss over in our analysis: transaction fees and slippage. For fixed-expiry futures trades that have a multi-percentage basis, the trading fees and slippage required to enter the trade are negligible (of course, unless the trader is pushing massive volume). This is not the case for perpetual funding rates, where the instantaneous spread between futures and spot prices are very small.

*The annualized perpetual funding rate for the BTC-PERP market on FTX. Funding rates are volatile and it is difficult to anticipate future funding rates with high certainty. Source: [FTX's API](https://docs.ftx.com/#overview).*

However, the no-arbitrage condition still holds for risk-neutral traders who have an expectation regarding future interest rates and perpetual funding rates. For instance, suppose a trader has \$100 USD and is choosing between a perpetual basis trade on asset `A` for a week or lending their USD for a week. If they expect the cumulative perpetual funding rate to be higher than the cumlative interest earned through USD loans for that week, by a margin higher than the transaction, then they would opt for the perpetual basis trade, buying `A` spot and selling `A`'s perp.

## Conclusion
In this second part of the *Basis Basics* series, we demonstrate a number of extensions to the simple basis trades given in Part 1. First, we discuss how leveraged basis trades can yield higher returns, but these returns come at the cost of increased basis risk. Second, we examine the relationship between fixed-rate loans and fixed-expiry basis trades, showing that they are similar but somewhat distinct, and give practical guidance on when to trade the basis rather than lend. Finally, we perform a similar analysis for perpetual futures, demonstrating their similarity to variable-interest loans and providing guidance on when to trade the funding rate rather than lend.

Basis trading is a unique form of trading, and although it can partially be replicated via lending, basis trading still has its own unique set of risk and return profiles. Traders should practice proper risk management when speculating on the basis, especially when doing so with leveraged positions.

[^1]: https://help.ftx.com/hc/en-us/articles/360053007671-Spot-Margin-Trading-Explainer
[^2]: https://compound.finance/documents/Compound.Whitepaper.pdf
[^3]: https://github.com/aave/aave-protocol/blob/master/docs/Aave_Protocol_Whitepaper_v1_0.pdf
[^4]: On-chain money market protocols allow borrowers' collateral to be lended out, and if the borrower's position becomes undercollateralized, their collateral can be seized. Here we assume that the lender is not using the lended funds as collateral, and thus there is no notion of liquidation risk for lenders.
[^5]: It is still important to note that we are assuming rational traders have infrastructure for entering and maintaining basis trades as easily as they do for fixed-rate loans, and that the venue on which they conduct basis trades is as low risk as the venue on which they conduct fixed-rate lending.
[^6]: https://en.wikipedia.org/wiki/Proportional_control